The so-called "religious" collegia of the Roman world, and the various Greek voluntary religious associations (e.g. ὀργειῶνες) are really not comparable to the church organization of the Christian world. For one thing, such associations were not really a "type." By that I mean that the voluntary associations that we call "religious" are called that because they satisfy the very broad conditions of being voluntary associations that have something to do with religion. In that respect yes, one might consider early church parishes as a sort of religious collegium, but such organizations did not refer to themselves with the same language and the resemblance is deceptive.

The Greek world knew several kinds of voluntary religious associations. By the Hellenistic Period the most important of these were probably the ὀργειῶνες and θίασοι. It's a matter of some dispute whether these associations existed before the fourth century and, if they did, whether they would have resembled the associations that existed in the later period. In a very broad sense, both types of associations were involved in the celebration of particular kinds of religious rituals, usually quite specific. However, the religious aspect of these associations should not be over-stressed such that we think that these are associations of worshipers, in the Christian sense, of the gods and heroes whose rituals they performed. It's not clear whether these associations were truly "standing," in that we can't really tell if they always had consistent membership from whose ranks the people who performed the rituals were drawn, or whether annually their membership was basically totally rebuilt. What we're quite certain of is that the actual religious activities of such associations were quite rare. Greek voluntary religious associations usually only performed a single ritual a year, typically for the hero's or god's most important procession or, if the god had multiple important festivals, for a particular procession or even a particular part of that ritual. And this is an important distinction. Early church communities met regularly for worship--an ἐκκλησία, while developing into the word for church in late antiquity, is an assembly. In this respect it might seem that there is continuity here. Greek voluntary religious associations, after all, held ἐκκλησίαι regularly. But these were modeled on the political meetings of the tribes and of the city community as a whole. Voluntary religious associations held ἐκκλησίαι κυρίαι (the "chief assembly," i.e. the first assembly of the month, an institution taken over from the tribes, demes, and the general Athenian assembly), and like the community as a whole passed decrees in their assemblies which were inscribed for internal use, in the same way that the city assembly passed decrees and displayed them publicly on stone stelae. We have quite a lot of these religious decrees, especially from Athens and the Piraeus, and while they often dealt with religious matters (e.g. the construction or extension of a shrine, or the appointment of a priest), the sense we get is that the association's main functions, except for the rare occasions when it was actually performing the rituals for which it was responsible, were basically administrative.

The situation is much the same for collegia cultorum and other similar religious collegia in the Roman world, for which our information is better but still lacking. Religious collegia similar to the Greek voluntary religious associations existed and were quite common--it seems likely that many of these continued to exist between the ban on most of the city's collegia in 64 and their revival under strict control sometime in the Principate. These sorts of collegia celebrated particular rituals or participated in larger, more elaborate rituals. But again, this was really only once a year typically, or after lengthy intervals. And these collegia resemble fairly closely the professional collegia, those of the butchers, the carpenters, and so forth. Collegia might be formed around the people who bore certain particular sacred objects during a particular annual festival, for example. What unites them is not their service to the god, but the particular function that they perform in service to the god. In this they probably also follow the collegia (in the institutional sense, not the sense of a voluntary association) of priests, which are distinguished by their highly professional, exclusive nature, which is intended to set them apart. In most "religious" collegia things were even weirder. Probably the most famous religious collegium--arguably the most famous collegium, period--is the collegium of worshipers (cultores) of Diana and Antinous. They are famous because their association's lex was preserved in an inscription (CIL 14.2112) that stands in the Museo Nazionale Romano--this is the only collegiate lex that survives, and indeed the only collegiate lex about whose very existence we have firm knowledge. The inscription--and the collegium--are very controversial, mainly because it's not clear how representative it is of anything else. However, what's certainly clear is that despite the name of the collegium, its members had pretty much nothing to do with the cult of Diana and Antinous. The cult and its rituals are mentioned in the lex a few times, but it's clearly not very important to the function of the collegium or the minds of its members. The collegium's primary purpose according to the lex was to provide monthly communal meals for its members and to ensure proper burial rights, for which a common fund was created to which members contributed monthly. This has created a lot of controversy--Mommsen thought that there existed a class of collegia funeraticia that arranged burials, which has been disproven by Andreas Bendlin--but it's obvious that even though there were regular meetings of this association (monthly, since collegia were not permitted in the Principate to meet more often than once a month) it was a social association. The so-called "cultores" were not meeting to worship or do anything particularly religious. The sense that we get is that this is not atypical. Collegia frequently had some sort of religious aspect to them, usually in the form of an annual ritual. Even professional collegia operated like this. The annual celebration of the Ludi Compitalicii, which was celebrated by the collegia in the city's vici, was shown by Flambard not to be the role of a class of collegia in the Republic (though this seems to have been the case in the Principate). Instead, this religious festival appears to have been celebrated in the Republic by the local professional collegia in the area, despite the fact that ostensibly at least many of these collegia would not have had any sort of official "religious" aspect in their titulature.

I should probably say something about mystery cults. I don't know very much about mystery cults honestly, especially in the later empire. In the Classical Period, however, mystery cults really don't work as an analog to the early church. Mystery cults were voluntary religious associations, yes, and unlike many other associations they were primarily religious. However, mystery cults were not as exclusive as they might seem. Mystery cults were, in the Classical Period, quite rare, and membership into them was a primarily community thing--a rather large portion of the population of Athens was part of the Eleusinian Mysteries. Mystery cults like that of Eleusis or of Isis were not regularly-meeting associations. Rather, they were associations to which one claimed membership, and with it certain special communal understanding with other members of the cult, but the actual religious activities were rare. Initiates of the Eleusinian Mysteries met exactly once a year. The situation was much the same for initiates of the cults of Isis, or Bacchus.